# Subtract simple interest from compound interest

Here's a question on compound interest and simple interest.

The difference in compound interest and simple interest for 2 years on a sum of money is \$160. If the simple interest for 2 years be \$2,880, the rate percent is ____ ?

How do I do it?

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Doesn't this depend on how frequently the interest is compounded? Annually? Monthly? Daily? Continuously? – MJD Dec 17 '12 at 15:25
@MJD the interest is compounded annually... – ShuklaSannidhya Dec 17 '12 at 15:38

Let's call the original principal $P$, and the interest rate $r$. Then the interest accrued in the first year is $Pr$.

If the interest is simple, the interest accrued in the second year is the same, $Pr$ again, for a total of $2Pr$. But if the interest is compound, the interest in the second year is on the original principal plus the first year's interest, $P + Pr$, and so is $(P + Pr)r = Pr + Pr^2$, rather than just $Pr$. The difference between the simple and compound interest accrued in two years is therefore $Pr^2$.

We are given that the simple interest is $2Pr = \$2,880$, and the difference between the two kinds of interest is$Pr^2 = \$160$. Dividing the second by the first gives:

$${ Pr^2\over 2Pr} = {160\over 2880} \\ \frac r2 = \frac1{18}\\ r = \frac19$$

Or if you prefer, 11.1%.

Then we can solve for the original principal $P$, and then check the values for $P$ and $r$ by calculating the simple and compound interest amounts on $P$ at rate $r$ to see if they match the givens in the question,

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How do we solve for 'P' ? – ShuklaSannidhya Dec 17 '12 at 16:57
How is the difference Pr^2. (Pr+Pr^2-2Pr= Pr^2-Pr)... Shouldn't it be Pr^2-Pr??? – ShuklaSannidhya Dec 17 '12 at 17:04
The simple interest in the first year is $Pr$, and in the second year $Pr$, for a total of $2Pr$. The compound interest in the first year is $Pr$, and in the second year $Pr + Pr^2$, for a total of $2Pr + Pr^2$. You solve for $P$ by taking $2Pr = \$2,880$and putting in the known value for$r$, leaving an equation in$P\$. – MJD Dec 17 '12 at 17:14
thanks a lot Mr. Dominus. – ShuklaSannidhya Dec 17 '12 at 17:24